The first thing Musar Abid did when he escaped the Islamic State this week was grab a razor.

“I shaved this morning,” he said, smiling and pointing to his cheeks, smooth for the first time since jihadist enforcers began requiring long beards two years ago. “I became a young man again.”

When Iraqi federal policemen taking part in the offensive to reclaim Mosul from the terrorist group approached his village, Mr. Abid, 41, was elated to see them coming.

For Mr. Abid, escape was an imperative — he had informed to the Iraqi police about conditions in the city, he said. “All of the world knows what life was like under Daesh,” he said, using the Arabic acronym for the Islamic State, which is also known as ISIS or ISIL.